T. Chantladze, N. Kandelaki, A. Lomtatidze, D. Ugulava

On a Generalization of the Dirichlet Integral

abstract:
Using the theory of spline functions, we investigate the problem of minimization of a generalized Dirichlet integral of the form
F_l(u) = Integral over W of (l² + Sum from i=1 to n of u_xi² ) ^p/2,    1 < p < infinity
where W is a bounded domain of an n-dimensional Euclidean space R_n, l >= 0 is a fixed number, and u_xi is a generalized derivative of the function u with respect to x_i according to Sobolev defined on W. Minimization is realized with respect to the functions u whose boundary values on G form a preassigned function, and for them F_l(u) is finite.